What is this idea of "Minimum Correlation"?

Question

So I was having a read of this paper here: Minimum correlation for any bivariate Geometric distribution. On the first page of he paper we encounter the following definition of "minimum correlation": $$ \rho_{-}\left(p_1,p_2\right) = \min\left\{\text{Corr}\left(X_1,X_2\right) : X_1\sim Geo\left(p_1\right), X_2\sim Geo\left(p_2\right)\right\} $$ For $p_1$ and $p_2$ on the unit interval (and apparently fixed). What I don't understand is how one can take a minimum of the correlation... Correlation, as I understand it, between two random variables should just be a scalar value, and not a function to be minimized.

Doing some more reading on the topic, I found this paper: Bivariate Geometric Distributions. At the end of section 2.3 we see a formula for the correlation:

$$ \text{Corr}\left(X_1,X_2\right) = \frac{-pq}{\left(1-r\right) \sqrt{(1-p)(1-q)}} $$ Where $X_1\sim Geo\left(p\right)$ and $X_2 \sim Geo\left(q\right)$ and $r = 1 - p- q$ (according to definition 2.1 in the paper above).

So I'm not sure what exactly is supposed to be done here with "minimum correlation". Can anyone explain what I'm missing?

Answer 1

The minimization is over the set $J(p_1,p_2)$ of all possible joint distributions $(X_1,X_2)$ for which $X_1$ and $X_2$ are discrete geometric distributions with parameters $p_1$ and $p_2$. Correlation of $X_1$ and $X_2$ is a function on $J$, and the minimum correlation is therefore a function of $(p_1,p_2)$. Calling that minimum correlation MinCor($p_1,p_2$), the subject of the paper is the function $M(p)=$ MinCor($p,p$).

Answer 2

I think what they mean in the paper is that you take the minimum over all correlations between two random variables that have geometric marginal distribution. To illustrate, the formula for the correlation is given by $$ \text{cor}(X,Y) = \frac{E(XY)-E(X)E(Y)}{\sqrt{\text{var}(X)\text{var}(Y)}}. $$ Now, all terms are dependent only on the marginal distribution except for $E(XY)$. The minimum thus minimizes over all bivariate distributions where the two marginals are both geometrically distributed. Compare for example the cases $X=Y$ and $X,Y$ independent.